David J. Green

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Cohomology of group number 2145 of order 128

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Ring generators

Ring relations

Completion information

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General information on the group

• The group has 4 minimal generators and exponent 16.
• It is non-abelian.
• It has p-Rank 3.
• Its center has rank 2.
• It has 2 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 3.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 3 and depth 2.
• The depth coincides with the Duflot bound.
• The Poincaré series is

  ( − 1) · (t6  +  t5  +  t2  +  t  +  1)

  (t  −  1)3 · (t2  +  1) · (t4  +  1)

• The a-invariants are -∞,-∞,-5,-3. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 7 minimal generators of maximal degree 8:

1. a_1_1, a nilpotent element of degree 1
2. b_1_0, an element of degree 1
3. b_1_2, an element of degree 1
4. c_1_3, a Duflot regular element of degree 1
5. a_5_27, a nilpotent element of degree 5
6. b_5_26, an element of degree 5
7. c_8_57, a Duflot regular element of degree 8

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Ring relations

There are 9 minimal relations of maximal degree 10:

1. a_1_1·b_1_0
2. b_1_0·b_1_2² + a_1_1³
3. a_1_1³·b_1_2²
4. b_1_0·a_5_27
5. a_1_1·b_5_26
6. b_1_2²·b_5_26 + a_1_1²·a_5_27
7. a_5_27·b_5_26
8. a_5_27² + a_1_1·b_1_2⁴·a_5_27 + a_1_1²·b_1_2³·a_5_27 + c_8_57·a_1_1²
9. b_5_26² + b_1_0⁵·b_5_26 + c_8_57·b_1_0²

About the group

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Data used for Benson′s test

• Benson′s completion test succeeded in degree 10.
• The completion test was perfect: It applied in the last degree in which a generator or relation was found.
• The following is a filter regular homogeneous system of parameters:
  1. c_1_3, a Duflot regular element of degree 1
  2. c_8_57, a Duflot regular element of degree 8
  3. b_1_2² + b_1_0², an element of degree 2
• The Raw Filter Degree Type of that HSOP is [-1, -1, 4, 8].
• The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].

About the group

Ring generators

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Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 2

1. a_1_1 → 0, an element of degree 1
2. b_1_0 → 0, an element of degree 1
3. b_1_2 → 0, an element of degree 1
4. c_1_3 → c_1_0, an element of degree 1
5. a_5_27 → 0, an element of degree 5
6. b_5_26 → 0, an element of degree 5
7. c_8_57 → c_1_1⁸, an element of degree 8

Restriction map to a maximal el. ab. subgp. of rank 3

1. a_1_1 → 0, an element of degree 1
2. b_1_0 → c_1_2, an element of degree 1
3. b_1_2 → 0, an element of degree 1
4. c_1_3 → c_1_0, an element of degree 1
5. a_5_27 → 0, an element of degree 5
6. b_5_26 → c_1_1²·c_1_2³ + c_1_1⁴·c_1_2, an element of degree 5
7. c_8_57 → c_1_1²·c_1_2⁶ + c_1_1⁸, an element of degree 8

Restriction map to a maximal el. ab. subgp. of rank 3

1. a_1_1 → 0, an element of degree 1
2. b_1_0 → 0, an element of degree 1
3. b_1_2 → c_1_2, an element of degree 1
4. c_1_3 → c_1_0, an element of degree 1
5. a_5_27 → 0, an element of degree 5
6. b_5_26 → 0, an element of degree 5
7. c_8_57 → c_1_1⁴·c_1_2⁴ + c_1_1⁸, an element of degree 8

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128